Messed up Armature Deformation
There are two problems with your mesh. First, your normals are inside out, which will give you some problems if you try to skin (or anything else, really). Go into edit mode, select all of your mesh, press ctrl-n. A good way to visualize the normals of a mesh is to turn off double-sided lighting. You can do this in the mesh data tab, where you find your vertex groups. Disable the option; if the mesh looks really light, the normals are facing the right way. If it looks dark, the normals are facing inward and need to be reversed. Your screenshots show inverted normals. 

Now you can rig the mesh. You already have certain bones set up to deform and not deform, which is good. 

First you need to delete your vertex groups on the mesh. This will remove any old weight data on the mesh so you can start clean and fresh. From here, just use automatic weighting. You will certainly need to fine-tune your weights after. 

The reason your mesh was deforming so terribly was because your weights were spread out over the whole mesh, not localized to the area with the respective bone. Leg bones were affecting mesh on the opposite side, torso bones had weight up in your arms, etc. 

This is a good overview of what weight painting is and how to do it. 

https://www.youtube.com/watch?v=S0VS0IVylzg

For documentation on weight painting, see the manual here: https://www.blender.org/manual/rigging/skinning/obdata.html